2012/03/27

Some Thoughts on Mathematics

It is a well-rehearsed complaint that the United States is "falling behind" the rest of the world in technical studies - meaning mathematics and science.  It is paradoxical, then, that the general United States curriculum features specific study of advanced mathematical topics - Algebra and Geometry.  Sometimes requirements have even advanced up to basic calculus or at least its underlying principles (indignified with the moniker "pre-calculus").  Most other countries teach Mathematics, perhaps glorified with a grade number, throughout middle and high school.

Speculating on the explanation for this oddity, I would suspect something like the following happened in American education: Years ago when "high school" would have been considered fairly advanced education, algebra and trigonometry and other advanced math would reasonably have been studied, possibly as electives; essential math would have been learned earlier.  Mathematics instruction likely suffered from the overall decline in American education over the (later part of the?) 20th century.  The language problems, I suppose, are documented better because writers felt them more critically.  Either the courses were never dropped in name, or people noticing the problems codified the "Algebra" and "Geometry" to be studied; however the rigor of the courses was slackened and the material diluted with the addition of other elements which were no longer being learned earlier - or which were part of newly expanding fields and felt to be "necessary".

As evidence, I have mainly the (I am tempted to say absurd) amount of review contained from year to year in the average modern US mathematics textbook series.  Admittedly mathematics is a subject I have some affinity for, but in my estimation the material generally spread over the three years from pre-algebra to second-year algebra either could be condensed into at most two years; or should be taught earlier; or is (at least from a conceptual standpoint) superfluous to the subject at hand.  An "Algebra II" class - I am speaking here from experience - by the end of the first month of classes can still be reviewing material theoretically learned up to three years before in "Pre-Algebra" (possibly even the first semester of that course) - linear equations.  True, the more advanced course has more detail and harder problems; I am not convinced that excuses the state of affairs.

I do not think - at this point in my career, at least - that the problem is the (nominal) focus of American courses.  The integrated approach adopted most famously by most Asian schools does have its advantages, mainly in maintaining a unity in the discipline; the focused study of a particular branch of a subject has off-setting advantages, mainly in ability to explain details rather than teaching by rote.  In disadvantages, the "subject-based" approach of American textbooks does tend to create artificial distinctions of one thing from another that ought to be known as related; however, an integrated approach obscures the focus achieved by distinguishing between the parts of a field of study.  Beyond that I am not qualified to comment - except to mention that more American textbooks seem to be actually using an essentially integrated approach, while maintaining their supposed subject matter in name only.

A more pressing problem is the study habits expected of students.  When a conscientious American parent wants the student to master a subject, and is willing to sacrifice grades if necessary, all is well and good.  When a lazy parent (to say nothing of the student or teachers) with "high expectations" wants the student to have an "A" but could care less about the subject, there is a problem - and the Asian parent determined that his child will have an A and prepared to expect that and make him work for it creates an advantage.  (I dislike relying on overplayed stereotypes, and can say from personal experience that - which should be obvious to any observer of human nature - not every Asian parent is in this regard an "Asian parent".  Some of them are quite prepared to look the other way on instance of, for instance, cheating; or even to berate teachers who attempt to discipline that behavior.  On a general comparison, however, the stereotype does hold true for fairly clear cultural reasons.  The comparison is therefore useful.  Also, I know next to nothing of European education, which is the other reasonable comparison point.)

So much for the problem.  What about a solution?  One obvious comment is that the best method of improvement must be increased expectations, even demands, on the part of parents and schools.  The effect of this simple change can be most clearly seen in modern America within the home- and classical-schooling community, where concerned parents created a demand - and have often been part of creating a supply - of improved, or at least diverse, curricula for language study both in English and the Classics.  Contrast this with mathematics education, especially in the public schools, where the number of available curricula in print has been steadily decreasing.  I believe there is a total of two courses remaining put out by large publishers, and one has much greater presence as best as I can ascertain.  My knee-jerk reaction is to blame Federal government meddling with standards, together with a human inclination to follow the Next Greatest Thing, but I could be wrong - the point is that with no serious competition and a largely captive market, innovation and diversity seems to be quickly becoming a matter of who has prettier pictures.

My own suggestions are limited by unfamiliarity with lower elementary curricula.  A makeshift solution I would propose as an upper school teacher would be to spend seventh (and maybe eighth?) grade focusing on practical math, starting wherever necessary and working up to whatever difficulty level is needed for scraping by (and hopefully more than scraping by) in life but without particular emphasis on unifying principles of "subjects"; the unifying principle of mathematics generally is (in my opinion - this would require an entire other essay) description and problem-solving.  With that foundation in place, you have a fourteen-year-old student who is capable of doing things like working out a simple budget (say, not overspending his allowance), working out basic problems in compound interest, calculating the price of a room's carpet, or not drawing to an inside straight (for reasons other than every book ever written with a card game in it saying so, not that that is a bad reason).

A high school can then teach those subjects which will be either necessary to a future career (as an engineer, architect, or the like) as electives, and require those useful to the formation of a thoughtful citizen ("Let no one ignorant of geometry...").  The question "how much math?" is a fascinating one I am not prepared to answer in the general case.  I am prepared to say that it would be easier to teach algebra thoroughly if a text did not interrupt - and perhaps need to interrupt - the fairly logical progression through the various variable functions with silliness about translations which should have been learned earlier, and bits of a trigonometry (for instance) which could either be learned later or cheerfully ignored.  Every calculus textbook I ever remember seeing not only confined itself politely to a strictly logical presentation of the calculus, but made assumptions of what the student knows.  If we imagine a calculus textbook written on the same principles as the average "Algebra 2", it would start by presenting methods to solve equations in one variable and would reteach the quadratic formula; in between types of derivatives there would be a discussion on graphing complex numbers.  An actual calculus text teaches a subject; I am not sure what the algebra book is trying to do.

1 comment:

  1. what is scientific notation
    Scientific Notation include in the mathematics course. In the world of science some time we deal with numbers which are very small and those which are very large. In some branches of science large numbers while in others very small numbers are used.

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